Open problems in the rigidity theory of actions with extreme deviation from multiple mixing

Tikhonov (2018) introduced $L^{(n)}$-systems, the class of measure-preserving actions exhibiting extremal deviation from multiple mixing — the asymptotic behavior of Ledrappier's counterexample to the weakened Rokhlin problem — and classified minimal totally ergodic non-Bernoulli mixing $L^{(n)}$-systems of $\mathbb{Z}^2$-actions by a single generating shape. The paper's closing remarks leave three open problems: (1) the rigidity theory covers only algebraic counterexamples to the weakened Rokhlin problem whose distinguished set has measure $1/2$ and which are governed by a single limit relation, whereas the source asserts the existence of algebraic counterexamples with a distinguished set of measure $1/3$, for which an analogous theory can be built from several simultaneous limit relations — whether a single limit relation can suffice for them is unclear; (2) the inverse problem of determining exactly which finite subsets of $\mathbb{Z}_+^2$ occur as generating shapes has only partial necessary conditions; (3) for the quantitative $\varepsilon$-relaxations ($L_\varepsilon^{(n)}$-systems), it is open whether one can choose an arbitrarily small $\varepsilon>0$ such that no $L_\varepsilon^{(n)}$ measure-preserving transformations exist — equivalently, since the classes are monotone in $\varepsilon$, whether a shape-independent threshold $\varepsilon_0>0$ exists below which no $L_\varepsilon^{(n)}$ transformation exists — beyond Tikhonov's own 2021 partial progress (a uniform, i.e. shape-independent, bound $\varepsilon\ge 1/16$ for multidimensional flows in the three-point case, and only shape-dependent bounds for transformations), the question remains open.

Background

By a measure-preserving action $L=\{L_g\}_{g\in G}$ of a topological group $G$ on the unit interval (with Lebesgue measure $\mu$) we mean a collection of invertible measure-preserving transformations satisfying $L_gL_h=L_{gh}$ such that $g\mapsto\mu(L_gA\cap B)$ is continuous for all measurable $A,B$; a sequence $g_i$ tends to infinity if it eventually leaves every bounded subset of $G$. The action is mixing if $\mu(L_{g_i}A\cap B)\to\mu(A)\mu(B)$ whenever $g_i\to\infty$. A family of $n$ sequences $\{T_i^{(1)}\},\dots,\{T_i^{(n)}\}$ drawn from the action is (fully) mixing if $\mu(\bigcap_j T_i^{(j)}A_j)\to\prod_j\mu(A_j)$ as $i\to\infty$, and $k$-fold mixing if every subfamily of $k$ sequences mixes.

Rokhlin's multiple-mixing problem (1949), still open, asks whether a mixing transformation is mixing of all orders, that is, whether mixing of higher order follows from $2$-fold mixing. The weakened Rokhlin problem, posed by Ryzhikov (1992), is wider: if a finite family of sequences of elements of an action is pairwise ($2$-fold) mixing, must it mix of all orders? No counterexample is known for transformations ($G=\mathbb{Z}$) or multidimensional flows ($G=\mathbb{R}^d$), but Ledrappier (1978) constructed the famous algebraic counterexample for $G=\mathbb{Z}^2$: a mixing action $\mathcal{L}$ by automorphisms of a compact abelian group, possessing a measurable set $A$ with $\mu(A)=1/2$ and $\mu(A\cap\mathcal{L}_{(1,0)}A\cap\mathcal{L}_{(0,1)}A)=0$ (a three-point relation), which forces $\mu(A\cap\mathcal{L}_{(2^k,0)}A\cap\mathcal{L}_{(0,2^k)}A)\to 0$ — a maximal possible deviation from $3$-fold mixing, for which the limit would have to be $1/8$. All known counterexamples to Rokhlin's problem are of this algebraic type.

Tikhonov (2015, 2018) axiomatized this extremal behavior. An action $L$ is an $L^{(n)}$-system if there exist a set $A$ with $0<\mu(A)<1$ and an $(n-1)$-fold mixing family of sequences $\{T_i^{(1)}\},\dots,\{T_i^{(n)}\}\subset L$ such that the measure of the symmetric difference $\mu(\Delta_j T_i^{(j)}A)$ tends to $0$ or to $1$ as $i\to\infty$; every $n$-point algebraic counterexample of Ledrappier type is an $L^{(n)}$-system. An $L^{(n)}$-system is minimal if $L$ is already the factor action on the sigma-algebra generated by all shifts $L_gA$. The 2018 paper proved: the measure of the distinguished set is always exactly $1/2$; there are no $L^{(n)}$-systems among transformations ($G=\mathbb{Z}$) and none among multidimensional flows ($G=\mathbb{R}^d$); and a shape — a finite set $H\subset G$ with $\mu(\Delta_{h\in H}L_hA)\in\{0,1\}$ — always exists. For minimal totally ergodic non-Bernoulli $L^{(n)}$-systems of $G=\mathbb{Z}^2$ (totally ergodic means every non-identity element of the action is ergodic; Bernoulli shifts, i.e. independent-symbol shift actions, are excluded), all shapes are generated by shifts and symmetric differences from a single generating shape, coincidence of generating shapes implies isomorphism of the actions, and Ledrappier's three-point example is the unique one with generating shape $\{(0,0),(1,0),(0,1)\}$ — a classification by a combinatorial invariant. Related algebraic-side information, in the different terminology of mixing shapes for algebraic $\mathbb{Z}^d$-actions on compact groups with Haar measure, is due to Ward (1997): every non-linear shape $I\subset\mathbb{Z}^d$ occurs as a minimal mixing shape of some algebraic action, and Einsiedler–Ward (2005) showed that different shapes give non-isomorphic zero-dimensional algebraic actions.

The quantitative relaxation studied in the source and its successor: an $L_\varepsilon^{(n)}$-system is defined as above except that the limit of $\mu(\Delta_j T_i^{(j)}A)$ need only be $\varepsilon$-close to $0$ or $1$. For abelian groups the existence of a mixing action failing $3$-fold mixing is equivalent to the existence of an $L_\varepsilon$-system with $\varepsilon<1/2$ (Tikhonov 2021), so quantitative bounds on achievable $\varepsilon$ measure how far from extremal any counterexample to multiple mixing can be.

Problem Statement

The common objective is to complete the rigidity theory of systems with (near-)extremal deviation from multiple mixing opened by Tikhonov's 2018 classification of minimal totally ergodic non-Bernoulli mixing $L^{(n)}$-systems. The source paper explicitly leaves three linked questions:

(1) Single limit relation for the measure-$1/3$ counterexamples. The 2018 theory covers algebraic counterexamples to the weakened Rokhlin problem whose distinguished set $A$ has $\mu(A)=1/2$ and whose rigidity is forced by one limit relation $\mu(\Delta_j T_i^{(j)}A)\to 0$ (or $1$); indeed, the source paper proves that every $L^{(n)}$-system has $\mu(A)=1/2$ (Assertion 2). Remark 4 of the source asserts the existence of algebraic counterexamples to the weakened Rokhlin problem with $\mu(A)=1/3$, without identifying them explicitly; these are not covered by the present single-relation theory, and for such examples an analogous theory can be built from several simultaneous limit relations. Question (posed, as in the source, at this programmatic level, and still open): is there a single limit relation under which an analogous rigidity theory for the measure-$1/3$ algebraic counterexamples can be based — that is, can one manage with one limit relation instead of several simultaneous ones?

(2) Classification of generating shapes. The classification invariant for minimal totally ergodic non-Bernoulli mixing $L^{(n)}$-systems of $\mathbb{Z}^2$ is a generating shape: a finite subset $H\subset\mathbb{Z}_+^2$ from which all shapes of the system are obtained by shifts and symmetric differences, in the sense of the classification theorem of Tikhonov (2018); here $\mathbb{Z}_+$ denotes the set of nonnegative integers. Determine exactly which finite subsets $H\subset\mathbb{Z}_+^2$ arise as generating shapes, i.e. occur for some minimal totally ergodic non-Bernoulli mixing $L^{(n)}$-system, with the order $n$ free. Known: a natural necessary condition is that $H$ contain elements with zero first coordinate and elements with zero second coordinate; this is not sufficient, since $H=\{(0,0),(0,1),(1,0),(1,1)\}$ is not a generating shape; Ward's algebraic mixing-shape results give related, not identical, information. A complete characterization (necessary and sufficient conditions) is requested.

(3) Uniform gap for near-extremal transformations. For $L_\varepsilon^{(n)}$-systems, where the limit of $\mu(\Delta_j T_i^{(j)}A)$ is only $\varepsilon$-close to $0$ or $1$: can one choose an arbitrarily small $\varepsilon>0$ such that no $L_\varepsilon^{(n)}$-transformations (i.e. $\mathbb{Z}$-actions) exist? Since the classes are monotone in $\varepsilon$, this is equivalent to asking for a threshold $\varepsilon_0>0$ below which no $L_\varepsilon^{(n)}$-transformation exists. As in the source (Remark 9), the order $(n)$ is left as a free superscript and the missing ingredient is shape-uniformity: a resolution is either a shape-independent positive lower bound $\varepsilon_0>0$, or an explicit construction of $L_\varepsilon^{(n)}$-transformations for arbitrarily small $\varepsilon>0$. Known: at $\varepsilon=0$ no such transformations exist (for every order $n$); for multidimensional flows the shape-independent bound $\varepsilon\ge 1/16$ holds in the three-point case (Tikhonov 2021, Corollary 3); for transformations only bounds depending on the shape are known — $\varepsilon\ge 1/(8(K(a,b)+1))$ with $K(a,b)\le 2^a-1$, in particular $\varepsilon\ge 2^{-\gcd(a,b)-3}$, for the three-point shape with exponents $a,b$ (Tikhonov 2021, Theorems 4–5) — so the existence of a shape-independent positive threshold is the open core.

These three questions jointly complete one program: (1) extends the theory's scope to all algebraic counterexamples to the weakened Rokhlin problem, (2) completes the classification invariant for the planar lattice, and (3) establishes the quantitative robustness of rigidity for transformations. The three parts are independent.

References

1. S. V. Tikhonov, "Rigidity of Actions with Extreme Deviation from Multiple Mixing", Mathematical Notes 103:6 (2018), 977-989; Russian original: Mat. Zametki 103:6 (2018), 912-926, DOI: 10.4213/mzm11743. DOI: 10.1134/S0001434618050346, https://doi.org/10.1134/S0001434618050346 (source paper; Russian full text at https://www.mathnet.ru/eng/mzm11743)
2. V. V. Ryzhikov, "Mixing, rank, and minimal self-joining of actions with an invariant measure", Sbornik: Mathematics 75:2 (1993), 405-427; Russian original: Mat. Sb. 183:3 (1992), 133-160. DOI: 10.1070/SM1993v075n02ABEH003391, https://doi.org/10.1070/SM1993v075n02ABEH003391
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11. S. V. Tikhonov, "Mixing Actions of Locally Compact Groups", Mathematical Notes 117:4 (2025), 617-628; Russian original: Mat. Zametki 117:4 (2025), 561-574. DOI: 10.1134/S0001434625030277, https://doi.org/10.1134/S0001434625030277